Fault-Tolerant Metric Dimension of Interconnection Networks

Sakander Hayat,Asad Khan,Muhammad Kamran Siddiqui,Muhammad Imran,Muhammad Yasir Hayat Malik

doi:10.1109/access.2020.3014883

Abstract

A fixed interconnection parallel architecture is characterized by a graph, with vertices corresponding to processing nodes and edges representing communication links. An ordered set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> of nodes in a graph <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is said to be a resolving set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> if every node in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is uniquely determined by its vector of distances to the nodes in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> . Each node in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> can be thought of as the site for a sonar or loran station, and each node location must be uniquely determined by its distances to the sites in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> . A fault-tolerant resolving set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> for which the failure of any single station at node location <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> leaves us with a set that still is a resolving set. The metric dimension (resp. fault-tolerant metric dimension) is the minimum cardinality of a resolving set (resp. fault-tolerant resolving set). In this article, we study the metric and fault-tolerant dimension of certain families of interconnection networks. In particular, we focus on the fault-tolerant metric dimension of the butterfly, the Benes and a family of honeycomb derived networks called the silicate networks. Our main results assert that three aforementioned families of interconnection have an unbounded fault-tolerant resolvability structures. We achieve that by determining certain maximal and minimal results on their fault-tolerant metric dimension.

Highlights

In order to understand the concept of metric and fault-tolerant metric dimension better, we provide an example of a tree T with matric dimension 10 and fault-tolerant metric dimension 14
Corollary 6.5: The family of silicate networks is a family of interconnection networks with an unbounded fault-tolerant metric dimension
Even though the fault-tolerant metric dimension problem might be polynomially solvable for Benes, butterfly and mesh networks, we believe that is NP-complete for bipartite graphs

Summary

INTRODUCTION

S. Hayat et al.: Fault-Tolerant Metric Dimension of Interconnection Networks this issue by adding the assumption that a faulty censor will not lead to the system failure as the remaining censors will still be able to deal with the intruder. Raza et al [31] studied applications of fault-tolerant metric dimension in convex polytopes. Raza et al [29] studied the extremal structure of graphs with respect to the fault-tolerant metric dimension. Fault-tolerant resolving set) in G is called the metric dimension β(G) For any set W ⊂ V (G), let γ (W ) be the set of all the common neighbors of vertices in W Based on this concept, Raza et al [31] determined the following interesting relation between a resolving and a fault-tolerant resolving set. W := ∪v∈W N [v] ∪ γ (N (v)) is a fault-tolerant resolving set of G

A TOPOLOGICAL REPRESENTATION OF BUTTERFLY AND BENES NETWORKS

COMPUTATIONAL AND ALGORITHMIC COMPLEXITY

CONCLUSION AND FUTURE WORK